Space vector modulation for matrix converter and current source converter

ABSTRACT

A converter includes a transformer including primary windings and secondary windings, switches connected to the primary windings, an output inductor connected to the secondary windings, and a controller connected to the switches. The controller turns the switches on and off based on dwell times calculated using space vector modulation with a reference current {right arrow over (I)} ref  whose magnitude changes with time.

BACKGROUND OF THE INVENTION 1. Field of the Invention

The present invention relates to space vector modulation (SVM). More specifically, the present invention relates to an improved SVM algorithm that can be used with matrix rectifiers, current-source rectifiers, and current-source inverters.

2. Description of the Related Art

FIG. 1 shows an isolated matrix rectifier, FIG. 2 shows a current-source rectifier, and FIG. 3 shows a current-source inverter. Each of the circuits shown in FIGS. 1-3 can be used either with known SVM methods discussed in this section or with the novel SVM methods according to the preferred embodiments of the present invention discussed in the Detailed Description of Preferred Embodiments section below.

In FIG. 1, “line side” refers to the portion of the circuit on the left-hand side of the transformer T_(r) that is connected to the line voltages u_(a), u_(b), u_(c) for each of the phases A, B, C, and “load side” refers to the portion of the circuit on the right-hand side of the transformer T_(r) that is connected to the output voltage u_(o), i.e., the load. On the line side, the three-phase AC current is combined into single-phase AC current, and on the load side, the single-phase AC current is rectified by diodes D₁ to D4 to provide DC current.

The isolated matrix rectifier includes a filter inductor L_(f) and a filter capacitor C_(f) that define a line-side filter that reduces the total harmonic distortion (THD), bi-directional switches S₁ to S₆ arranged in a bridge as a 3-phase-to-1-phase matrix converter, a transformer T_(r) that provides high-voltage isolation between the line-side circuit and the load-side circuit, four diodes D₁ to D4 arranged in a bridge to provide output rectification, an output inductor L_(o), and an output capacitor C_(o) that define a filter for the output voltage. Bi-directional switches are used in this isolated matrix rectifier to open or close the current path in either direction. As shown in FIG. 1, the bi-directional switch includes two uni-directional switches connected in parallel.

THD is defined as the ratio of the RMS amplitude of the higher harmonic frequencies to the RMS amplitude of the fundamental frequency:

$\begin{matrix} {{THD} = \frac{\sqrt{\Sigma_{k = 2}^{\infty}V_{k}^{2}}}{V_{1}}} & (1) \end{matrix}$

where V₁ is the amplitude of the fundamental frequency and V_(k) is the amplitude of the higher harmonic frequencies. It is desirable to reduce the THD because the harmonic current can be injected back into the power system.

SVM is an algorithm for the pulse-width modulation (PWM) of the bi-directional switches S₁ to S₆. That is, SVM is used to determine when the bi-directional switches S₁ to S₆ should be turned on and off. The bi-directional switches S₁ to S₆ are controlled by digital signals, e.g., either one or zero. Typically, a one means the switch is on, and a zero means the switch is off. In PWM, the width of the on signal, which controls how long a switch is turned on, is modulated, or changed.

In known SVM, the main assumption is that the DC current is constant, which requires that the load-side inductor L_(o) should be infinite in theory and that the power converter should be only used in continuous-conduction mode (CCM) operation. CCM occurs when the current through the load-side inductor L₀ is always above zero. In contrast to CCM, discontinuous-conduction mode (DCM) occurs when the current through the load-side inductor L_(o) can be zero. The problem with using known SVM with DCM is large THD, as shown in FIG. 10E. It is impossible to provide an infinite load-side inductor L_(o) in practice. Although it is possible to provide a load-side inductor L_(o) with a very large inductance, doing so requires providing a large inductor that makes design difficult. It is impractical to assume that a power converter will only be used in CCM operation in any application that includes light-load conditions in which the power converter can be in DCM operation.

For the isolated matrix rectifier shown in FIG. 1, a switching function S_(i) can be defined as:

$\begin{matrix} {S_{i} = \left\{ {{\begin{matrix} {1,{S_{i}\mspace{14mu} {turn}\mspace{14mu} {on}}} \\ {0,{S_{i}\mspace{14mu} {turn}\mspace{14mu} {off}}} \end{matrix}i} \in \left\{ {1,2,3,4,5,6} \right\}} \right.} & (2) \end{matrix}$

where S_(i) is the switching function for the i^(th) switch. For example, if S₁=1, then switch S₁ is on, and if S₁=0, then switch S₁ is off.

Only two switches can be turned on at the same time to define a single current path. For example, if switches S₁ and S₆ are on, a single current path is defined between phases A and B through the transformer T_(r). If only two switches can conduct at the same time, with one switch in the top half of the bridge (S₁, S₃, S₅) and with the other switch in the bottom half of the bridge (S₂, S₄, S₆), then there are nine possible switching states as listed in Tables 1 and 2, including six active switching states and three zero switching states. In Table 1, line currents i_(a), i_(b), i_(c) are the currents in phases A, B, C, and the line-side current i_(p) is the current through the primary winding of the transformer T_(r). In Table 2, the transformer turns ratio k is assumed to be 1 so that the inductor current i_(L) is equal to the line-side current i_(p).

TABLE 1 Space Vectors, Switching States, and Phase Currents Space Switching States Vector S₁ S₂ S₃ S₄ S₅ S₆ i_(a) i_(b) i_(c) I₁ 1 0 0 0 0 1 i_(p) −i_(p) 0 I₂ 1 1 0 0 0 0 i_(p) 0 −i_(p) I₃ 0 1 1 0 0 0 0 i_(p) −i_(p) I₄ 0 0 1 1 0 0 −i_(p) i_(p) 0 I₅ 0 0 0 1 1 0 −i_(p) 0 i_(p) I₆ 0 0 0 0 1 1 0 −i_(p) i_(p) I₇ 1 0 0 1 0 0 0 0 0 I₈ 0 0 1 0 0 1 0 0 0 I₉ 0 1 0 0 1 0 0 0 0

TABLE 2 Space Vectors, Switching States, and Phase Currents Space Switching States Vector S₁ S₂ S₃ S₄ S₅ S₆ i_(a) i_(b) i_(c) I₁ 1 0 0 0 0 1 i_(L) −i_(L) 0 I₂ 1 1 0 0 0 0 i_(L) 0 −i_(L) I₃ 0 1 1 0 0 0 0 i_(L) −i_(L) I₄ 0 0 1 1 0 0 −i_(L) i_(L) 0 I₅ 0 0 0 1 1 0 −i_(L) 0 i_(L) I₆ 0 0 0 0 1 1 0 −i_(L) i_(L) I₇ 1 0 0 1 0 0 0 0 0 I₈ 0 0 1 0 0 1 0 0 0 I₉ 0 1 0 0 1 0 0 0 0

The active and zero switching states can be represented by active and zero vectors. A vector diagram is shown in FIG. 4, with the six active vectors {right arrow over (I)}₁˜{right arrow over (I)}₆ and the three zero vectors {right arrow over (I)}₇˜{right arrow over (I)}₉ . The active vectors {right arrow over (I)}₁˜{right arrow over (I)}₆ , form a regular hexagon with six equal sectors I-VI, and the zero vectors {right arrow over (I)}₇˜{right arrow over (I)}₉ lie at the center of the hexagon.

The relationship between the vectors and the switching states can be derived as follows.

Because the three phases A, B, C are balanced:

i _(a)(t)+i _(b)(t)+i _(c)(t)=0   (3)

where i_(a)(t), i_(b)(t), and i_(c)(t) are the instantaneous currents in the phases A, B, and C. Using equation (3), the three-phase currents i_(a)(t), i_(b)(t), and i_(c)(t) can be transformed into two-phase currents in the α−β plane using the following transformation:

$\begin{matrix} {\begin{bmatrix} {i_{\alpha}(t)} \\ {i_{\beta}(t)} \end{bmatrix} = {{\frac{2}{3}\begin{bmatrix} 1 & {- \frac{1}{2}} & {- \frac{1}{2}} \\ 0 & {- \frac{\sqrt{3}}{2}} & {- \frac{\sqrt{3}}{2}} \end{bmatrix}}\begin{bmatrix} {i_{a}(t)} \\ {i_{b}(t)} \\ {i_{c}(t)} \end{bmatrix}}} & (4) \end{matrix}$

where i_(α)(t), i_(β)(t) are the instantaneous currents in the phases α, β. A current vector I(t) can be expressed in the α−β plane as:

{right arrow over (I)}(t)=i _(α)(t)+ji _(β)(t)   (5)

{right arrow over (I)}(t)=⅔[i _(a)(t)e ^(j0) +i _(b)(t)e ^(j2π/3) +i _(c)(t)e ^(j4π/3)]  (6)

where j is the imaginary number and e^(jx)=cos x+j sin x. Then the active vectors in FIG. 4 are provided by:

$\begin{matrix} {{\overset{\rightarrow}{I_{k}} = {{\frac{2}{\sqrt{3}}\frac{I_{d}}{k}e^{j{({{{({k - 1})}\frac{\pi}{3}} - \frac{\pi}{6}})}}\mspace{14mu} {for}\mspace{14mu} k} = 1}},2,\ldots,6} & (7) \end{matrix}$

The isolated matrix rectifier's controller determines a reference current {right arrow over (I)}_(ref) and calculates the on and off times of the switches S₁ to S₆ to approximate the reference current {right arrow over (I)}_(ref) to produce the line-side current i_(a) and i_(b). The reference current {right arrow over (I)}_(ref) preferably is sinusoidal with a fixed frequency and a fixed magnitude: {right arrow over (I)}_(ref)=I_(ref)e^(jθ). The fixed frequency is preferably the same as the fixed frequency of each of the three-phase i_(a)(t), i_(b)(t), and i_(c)(t) to reduce harmful reflections. The controller determines the magnitude of the reference current {right arrow over (I)}_(ref) to achieve a desired output voltage u_(o). That is, the controller can regulate the output voltage u_(o) by varying the magnitude of the reference current {right arrow over (I)}_(ref).

The reference current {right arrow over (I)}_(ref) moves through the a-B plane. The angle θ is defined as the angle between the α-axis and the reference current {right arrow over (I)}_(ref). Thus, as the angle θ changes, the reference current {right arrow over (I)}_(ref) sweeps through the different sectors.

The reference current {right arrow over (I)}_(ref) can be synthesized by using combinations of the active and zero vectors. Synthesized means that the reference current {right arrow over (I)}_(ref) can be represented as a combination of the active and zero vectors. The active and zero vectors are stationary and do not move in the α−β plane as shown in FIG. 4. The vectors used to synthesize the reference current {right arrow over (I)}_(ref) change depending on which sector the reference current {right arrow over (I)}_(ref) is located. The active vectors are chosen by the active vectors defining the sector. The zero vector is chosen for each sector by determining which on switch the two active vectors have in common and choosing the zero vector that also includes the same on switch. Using the zero vectors allows the magnitude of the line-side current i_(p) to be adjusted.

For example, consider when the current reference {right arrow over (I)}_(ref) is in sector I. The active vectors {right arrow over (I)}₁ and {right arrow over (I)}₂ define sector I. The switch S₁ is on for both active vectors {right arrow over (I)}₁ and {right arrow over (I)}₂. The zero vector {right arrow over (I)}₇ also has the switch S₁ on. Thus, when the reference current {right arrow over (I)}_(ref) is located in sector I, the active vectors {right arrow over (I)}₁ and {right arrow over (I)}₂ and zero vector {right arrow over (I)}₇ are used to synthesize the reference current {right arrow over (I)}_(ref), which provides the following equation, with the right-hand side of the equation resulting from vector {right arrow over (I)}₇ being a zero vector with zero magnitude:

$\begin{matrix} {{\overset{\rightarrow}{I}}_{ref} = {{{\frac{T_{1}}{T_{s}}{\overset{\rightarrow}{I}}_{1}} + {\frac{T_{2}}{T_{s}}{\overset{\rightarrow}{I}}_{2}} + {\frac{T_{7}}{T_{s}}{\overset{\rightarrow}{I}}_{7}}} = {{\frac{T_{1}}{T_{s}}{\overset{\rightarrow}{I}}_{1}} + {\frac{T_{2}}{T_{s}}{\overset{\rightarrow}{I}}_{2}}}}} & (8) \end{matrix}$

where T₁, T₂, and T₀ are the dwell times for the corresponding active switches and T_(s) is the sampling period.

The dwell time is the on time of the corresponding switches. For example, T₁ is the on time of the switches S₁ and S₆ for the active vector I₁. Because switch S₁ is on for each of vectors {right arrow over (I)}₁, {right arrow over (I)}₂, and {right arrow over (I)}₇, the switch S₁ is on the entire sampling period T_(s). The ratio T₁/T_(s) is the duty cycle for the switch S₆ during the sampling period T_(s).

The sampling period T_(s) is typically chosen such that the reference current {right arrow over (I)}_(ref) is synthesized multiple times per sector. For example, the reference current {right arrow over (I)}_(ref) can be synthesized twice per sector so that the reference current {right arrow over (I)}_(ref) is synthesized twelve times per cycle, where one complete cycle is when the reference current {right arrow over (I)}_(ref) goes through sectors I-VI.

The dwell times can be calculated using the ampere-second balancing principle, i.e., the product of the reference current {right arrow over (I)}_(ref) and sampling period T_(s) equals the sum of the current vectors multiplied by the time interval of synthesizing space vectors. Assuming that the sampling period T_(s) is sufficiently small, the reference current {right arrow over (I)}_(ref) can be considered constant during sampling period T_(s). The reference current {right arrow over (I)}_(ref) can be synthesized by two adjacent active vectors and a zero vector. For example, when the reference current {right arrow over (I)}_(ref) is in sector I as shown in FIG. 5, the reference current {right arrow over (I)}_(ref) can be synthesized by vectors {right arrow over (I)}₁, {right arrow over (I)}₂, and {right arrow over (I)}₇. The ampere-second balancing equation is thus given by the following equations:

{right arrow over (I)} _(ref) T _(s) ={right arrow over (I)} ₁ T ₁ +{right arrow over (I)} ₂ T ₂ +{right arrow over (I)} ₇ T ₇   (9)

T _(s) =T ₁ +T ₂ +T ₇   (10)

where T₁, T₂, and T₇ are the dwell times for the vectors {right arrow over (I)}₁, {right arrow over (I)}₂, and {right arrow over (I)}₇ and T_(s) is sampling time. Then the dwell times are given by:

$\begin{matrix} {T_{1} = {{mT}_{s}\mspace{14mu} {\sin \left( {{\pi \text{/}6} - \theta} \right)}}} & (11) \\ {T_{2} = {{mT}_{s}\mspace{14mu} {\sin \left( {{\pi \text{/}6} + \theta} \right)}}} & (12) \\ {{T_{7} = {T_{s} - T_{1} - T_{2}}}{where}} & (13) \\ {{m = {k\frac{I_{ref}}{i_{L}}}},} & (14) \end{matrix}$

θ is sector angle between current reference {right arrow over (I)}_(ref), and α-axis shown in FIG. 5, and k is the transformer turns ratio.

However, the above dwell time calculations are based on the assumption that the inductor current i_(L) is constant. If the inductor current i_(L) has ripples, the dwell time calculation based on these equations is not accurate. The larger the ripple, the larger the error will be. As a result, the THD of the line-side current will be increased. In actual applications, the load-side inductance is not infinite, and the current ripple always exists. As shown in FIG. 6A, if the load-side inductance is small, then the current ripple is too large to use known SVM. As shown in FIG. 6B, to provide acceptable waveforms and to reduce line-side THD, the load-side inductance must be very large to reduce the current ripple and to come as close as possible to a theoretical value.

Known SVM can also be applied to the current-source rectifier in FIG. 2 and to the current-source inverter in FIG. 3 using the same techniques as discussed above with respect to equations (9)-(14).

A large load-side inductance has the problems of large size, excessive weight, and high loss, for example. The current ripple in a practical inductor also has the problems in modulation signals using traditional SVM, including increased line-side THD. In addition, DCM is unavoidable when the load varies. Under light loads, the load-side inductor L_(o) might be in DCM without a dummy load.

In known SVM for matrix rectifiers, current-source rectifiers, and current-source inverters, the DC current is assumed to be constant or the current ripple is assumed to be very small. Thus, known SVM includes at least the following problems:

-   -   1) The Load-side inductance must be large to maintain small         current ripple.     -   2) As a result of 1) the Load-side inductor size must be large.     -   3) Current ripple increases the THD of the line-side current.     -   4) Line-side current THD is high at light load.     -   5) Known SVM can only be used in CCM operation.

SUMMARY OF THE INVENTION

To overcome the problems described above, preferred embodiments of the present invention provide an improved SVM with the following benefits:

-   -   1) Reduced load-side inductance.     -   2) Reduced load-side inductor size.     -   3) Decreased THD of the line-side current, even with large         current ripple or light-load condition.     -   4) Improved SVM is capable of being used with both DCM and CCM         modes.     -   5) Improved SVM is simple and is capable of being calculated in         real time.

A preferred embodiment of the present invention provides a converter that includes a transformer including primary windings and secondary windings, switches connected to the primary windings, an output inductor connected to the secondary windings, and a controller connected to the switches. The controller turns the switches on and off based on dwell times calculated using space vector modulation with a reference current {right arrow over (I)}_(ref) whose magnitude changes with time.

Another preferred embodiment of the present invention provides a corresponding space-vector-modulation method.

Preferably, the switches include six switches; the space vector modulation includes using six active switching states and three zero switching states; a current space is divided into six sectors by the six active switching states such that a vector with θ=0 is located halfway between two of the active switching states; and magnitudes of the six active switching states change with time.

The controller preferably turns the six switches on and off based on dwell times that are calculated based on an ampere-second balance equation:

{right arrow over (I)}I _(ref) T _(s)=∫₀ ^(T) ^(α) {right arrow over (I)} _(α) dt+∫ ₀ ^(T) ^(β) {right arrow over (I)} _(β) dt+∫ ₀ ^(T) ^(o) {right arrow over (I)} ₀ dt

where {right arrow over (I)}_(ref)=I_(ref) e^(jθ), θ is an angle between the reference current {right arrow over (I)}_(ref) and the vector with θ=0, T_(s) is a sampling period, {right arrow over (I)}_(α), {right arrow over (I)}_(β), {right arrow over (I)}₀, are three nearest adjacent active vectors to {right arrow over (I)}_(ref) , and T_(α),T_(β),T₀ are dwell times of {right arrow over (I)}_(α),{right arrow over (I)}_(β),{right arrow over (I)}₀. The controller preferably turns the six switches on and off based on a vector sequence {right arrow over (I)}_(α), {right arrow over (I)}₀, −{right arrow over (I)}_(β), {right arrow over (I)}₀, {right arrow over (I)}_(β), {right arrow over (I)}₀, −{right arrow over (I)}_(α), {right arrow over (I)}₀, during the sampling period T_(s). The controller preferably turns the six switches on and off based on a timing sequence T_(α)/2, T₀/4, T_(β)/2,T₀/4, T_(β)/2, T₀/4,T_(α)/2,T₀/4, during the sampling period T_(s).

Preferably, the controller calculates the dwell times using:

$T_{\alpha} = \frac{{- B} + \sqrt{B^{2} + {{AC}\mspace{14mu} {\sin \left( {{\pi \text{/}6} - \theta} \right)}}}}{A}$ $T_{\beta} = {\frac{{- B} + \sqrt{B^{2} + {{AC}\mspace{14mu} {\sin \left( {{\pi \text{/}6} - \theta} \right)}}}}{A} \cdot \frac{\sin \left( {{\pi \text{/}6} + \theta} \right)}{\sin \left( {{\pi \text{/}6} - \theta} \right)}}$ T₀ = T_(s) − T_(α) − T_(β) where $A = {\left( {{4u_{1\alpha}\text{/}k} - u_{o}} \right) + {\left( {{4u_{1\beta}\text{/}k} - u_{o}} \right)\frac{\sin \left( {{\pi \text{/}6} + \theta} \right)}{\sin \left( {{\pi \text{/}6} - \theta} \right)}}}$ B = 4L_(o)I_(L 0) − 3u_(o)T_(s)/2 C = 8kL_(o)I_(ref)T_(s)

u_(1α) is a line-to-line voltage depending on the active switching state {right arrow over (I)}_(α), u_(1β) is a line-to-line voltage depending on the active switching state {right arrow over (I)}_(β), k is a transformer turns ratio, u_(o) is an output voltage of the converter, θ is the angle between the reference current {right arrow over (I)}_(ref) and the vector with θ=0, L_(o) is an inductance of the output inductor, I_(L0) is the current through inductor L_(o) at a beginning of the sampling period T_(s), T_(s) is the sampling period, and I_(ref) is a magnitude of the vector {right arrow over (I)}_(ref).

The controller preferably calculates the dwell times using:

$T_{\alpha} = {2\sqrt{\frac{{kI}_{ref}L_{o}T_{s}{\sin \left( {{\pi \text{/}6} - \theta} \right)}}{{u_{1\; \alpha}\text{/}k} - u_{o}}}}$ $T_{\beta} = {2\sqrt{\frac{{kL}_{o}I_{ref}T_{s}{\sin \left( {{\pi \text{/}6} + \theta} \right)}}{{u_{1\; \beta}\text{/}k} - \mu_{o}}}}$

where k is a transformer turns ratio, L_(o) is an inductance of the output inductor, I_(ref) is the magnitude of the vector {right arrow over (I)}_(ref), T_(s) is the sampling period, θ is an angle between the reference current {right arrow over (I)}_(ref) and the vector with θ=0, u_(1α) is a line-to-line voltage depending on the active switching state {right arrow over (I)}_(α), u_(1β) is a line-to-line voltage depending on the active switching state {right arrow over (I)}_(β), and u_(o) is an output voltage of the converter.

The converter is preferably one of a matrix rectifier, a current-source rectifier, and a current-source inverter. The converter is preferably operated in a continuous-conduction mode or a discontinuous-conduction mode.

The above and other features, elements, characteristics, steps, and advantages of the present invention will become more apparent from the following detailed description of preferred embodiments of the present invention with reference to the attached drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a circuit diagram of an isolated matrix rectifier.

FIG. 2 is a circuit diagram of a current-source rectifier.

FIG. 3 is a circuit diagram of a current-source inverter.

FIG. 4 shows a current-space vector hexagon.

FIG. 5 shoes the synthesis of reference current {right arrow over (I)}_(ref) using I₁ and I₂ using known SVM.

FIGS. 6A and 6B show ideal and real DC current waveforms.

FIG. 7 shows the synthesis of reference current {right arrow over (I)}_(ref) using I_(α) and I_(β) using SVM of a preferred embodiment of the present invention.

FIG. 8 shows the waveforms of the isolated matrix rectifier shown in FIG. 1.

FIGS. 9A, 9C, and 9E show waveforms of the isolated matrix rectifier shown in FIG. 1 in CCM using known SVM, and FIGS. 9B, 9D, and 9F show corresponding waveforms of the isolated matrix rectifier shown in FIG. 1 in CCM using SVM according to various preferred embodiments of the present invention.

FIGS. 10A, 10C, and 10E show waveforms of the isolated matrix rectifier shown in FIG. 1 in DCM using known SVM, and FIGS. 10B, 10D, and 10F show corresponding waveforms of the isolated matrix rectifier shown in FIG. 1 in DCM using SVM according to various preferred embodiments of the present invention.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

Preferred embodiments of the present invention improve the known SVM. The improved SVM is capable of being used with both DCM and CCM operation, is capable of being used with smaller load-side inductors, and reduces line-side THD.

As with the known SVM, the improved SVM includes nine switching states, including six active switching states and three zero switching states as shown in FIG. 4, that are used to synthesize the reference current {right arrow over (I)}_(ref) as shown in FIG. 7. However, in the improved SVM, the six active switching states, although stationary, are assumed to change with time. That is, the magnitude of the active switching states changes with time which is true in actual application.

The reference current {right arrow over (I)}_(ref) preferably is synthesized by the three nearest vectors {right arrow over (I)}_(α), {right arrow over (I)}_(β), {right arrow over (I)}₀ as shown in FIG. 7, and the dwell time of each vector is T_(α),T_(β),T₀. Here, (α, β) represent the subscript of the pair of active vectors in each sector such as (1,2) or (2,3) or (3,4) or (5,6) or (6,1). The dwell times preferably are calculated based on the principle of ampere-second balance. Because of current ripple, the inductor current is not constant, so the ampere-second balance of equation (9) becomes:

{right arrow over (I)} _(ref)T_(s)=∫₀ ^(T) ^(α) {right arrow over (I)} _(α) dt+∫ ₀ ^(T) ^(β) {right arrow over (I)} _(β) dt+∫ ₀ ^(T) ⁰ {right arrow over (I)} ₀ dt   (15)

Applying equation (15) to the isolated matrix rectifier shown in FIG. 1, provides the following analysis. The following assumptions are made in the following analysis:

-   -   1) Transformer T_(r) is ideal; and     -   2) In one sampling period T_(s), phase voltages u_(a), u_(b),         u_(c) are constant.

Because of the isolation provided by the transformer, the output voltage of the matrix converter u₁(t) must alternate between positive and negative with high frequency to maintain volt-sec balance. Thus, the preferred vector sequence in every sampling period T_(s) is divided into eight segments as İ_(α), İ₀, −{right arrow over (I)}_(β), İ₀, {right arrow over (I)}_(β), İ₀, −İ_(α), İ₀, and the dwell time of each vector is respectively T_(α)/2, T₀/4, T _(β)/2,T₀/4, T_(β)/2, T₀/4,T_(α)/2, T₀/4. However, the sequence of the active vectors and zero vectors can be combined in different ways, and the dwell time for the zero vectors is not necessary to be divided equally. For example, the vector sequence could be six segments as {right arrow over (I)}_(α), {right arrow over (I)}_(β), {right arrow over (I)}₀, −{right arrow over (I)}_(β), {right arrow over (I)}₀, with dwell time T_(α)/2, T_(β)/2, T₀/2, T_(α)/2, T_(β)/2, T₀/2, respectively. Only the case with eight segments as {right arrow over (I)}_(α), {right arrow over (I)}₀, −{right arrow over (I)}_(β), {right arrow over (I)}₀, {right arrow over (I)}_(β), {right arrow over (I)}₀, −{right arrow over (I)}_(α), {right arrow over (I)}₀, and the dwell time of each vector with T_(α)/2, T₀/4, T_(β)/2,T₀/4, T_(β)/2, T₀/4,T_(α)/2, T₀/4 is used as an example to show how the dwell times can be calculated to eliminate the effect of the current ripple on load side. FIG. 8 shows the waveforms of the matrix converter output voltage u₁(t) the inductor current i_(L)(t), the matrix converter output current i_(p)(t), and the phase current i_(a)(t). The inductor current i_(L)(t) at the time t₀, t₁, t₂, t₃, t₄, t₅, t₆, t₇, t₈, where t₁ and (t₇—t₆)=T_(α)/2, (t₃−t₂) and (t₅−t₄)=T_(β)/2, and the dwell time of the zero vectors are all T₀/4, can be described in equation (16):

$\begin{matrix} {{I_{Li} = {{I_{{Li} - 1} + {{\frac{u_{Li}}{L_{o}} \cdot \left( {t_{i} - t_{i - 1}} \right)}\mspace{14mu} i}} = 1}},2,3,4,5,6,7,8} & (16) \end{matrix}$

where the u_(Li) is the voltage of load-side inductor between times t_(i−1) and t_(i) and L_(o) is the inductance of the load-side inductor L_(o). The instantaneous value of the load-side inductor current is provided by:

$\begin{matrix} {{i_{L}(t)} = {{I_{{Li} - 1} + {{\frac{u_{Li}}{L_{o}} \cdot \left( {t - t_{i - 1}} \right)}\mspace{14mu} t_{i - 1}}} < t < t_{i}}} & (17) \end{matrix}$

The output current i_(p) of the matrix converter is provided by:

$\begin{matrix} {{i_{p}(t)} = \left\{ {\begin{matrix} {{{i_{L}(t)}\text{/}k}\mspace{14mu}} & {u_{1} > 0} \\ {{- {i_{L}(t)}}\text{/}k} & {u_{1} < 0} \end{matrix} = {{{gi}_{L}(t)}\text{/}k}} \right.} & (18) \end{matrix}$

where k is turns ratio of the transformer and the sign function g is defined by:

$\begin{matrix} {g = \left\{ {\begin{matrix} {1\mspace{14mu}} & {u_{1} > 0} \\ {- 1} & {u_{1} < 0} \end{matrix},} \right.} & (19) \end{matrix}$

Using equation (18), equation (7) for the active vectors becomes:

{right arrow over (I)}_(k)2/√{square root over (3)}i _(p)(t)e ^(j((k−1)π/3−π/6)) k−1,2,3,4,5,6   (20)

Substituting equations (17), (18), and (20) into the ampere-second balancing equation (15) provides:

$\begin{matrix} \begin{matrix} {{{\overset{\rightharpoonup}{I}}_{ref}T_{s}} = {{\int_{0}^{t_{1}}{{\overset{\rightharpoonup}{I}}_{\alpha +}{dt}}} + {\int_{t_{2}}^{t_{3}}{{\overset{\rightharpoonup}{I}}_{\beta -}\ {dt}}} + {\int_{t_{4}}^{t_{5}}{{\overset{\rightharpoonup}{I}}_{\beta +}{dt}}} + {\int_{t_{6}}^{t_{7}}{{\overset{\rightharpoonup}{I}}_{\alpha -}{dt}}}}} \\ {= {{\int_{0}^{T_{\alpha}\text{/}2}{{\overset{\rightharpoonup}{I}}_{\alpha +}{dt}}} + {\int_{0}^{T_{\beta}\text{/}2}{{\overset{\rightharpoonup}{I}}_{\beta -}{dt}}} + {\int_{0}^{T_{\beta}\text{/}2}{{\overset{\rightharpoonup}{I}}_{\beta +}{dt}}} + {\int_{0}^{T_{\alpha}\text{/}2}{{\overset{\rightharpoonup}{I}}_{\alpha -}{dt}}}}} \\ {= {{\int_{0}^{T_{\alpha}\text{/}2}{{\frac{2}{\sqrt{3}} \cdot \frac{1}{k}}\left( {I_{L\; 0} + {\frac{u_{L\; 1}}{L_{o}}t}} \right)e^{j{({{{\alpha\pi}\text{/}3} - {\pi \text{/}6}})}}{dt}}} + {\int_{0}^{T_{\beta}\text{/}2}{{\frac{2}{\sqrt{3}} \cdot \frac{1}{k}}\left( {I_{L\; 2} + {\frac{u_{L\; 3}}{L_{o}}t}} \right)e^{j{({{{\beta\pi}\text{/}3} - {\pi \text{/}6}})}}{dt}}} +}} \\ {{{\int_{0}^{T_{\beta}\text{/}2}{{\frac{2}{\sqrt{3}} \cdot \frac{1}{k}}\left( {I_{L\; 4} + {\frac{u_{L\; 5}}{L_{o}}t}} \right)e^{j{({{{\beta\pi}\text{/}3} - {\pi \text{/}6}})}}{dt}}} + {\int_{0}^{T_{\alpha}\text{/}2}{{\frac{2}{\sqrt{3}} \cdot \frac{1}{k}}\left( {I_{L\; 6} + {\frac{u_{L\; 7}}{L_{o}}t}} \right)e^{j{({{{\beta\pi}\text{/}3} - {\pi \text{/}6}})}}{dt}}}}} \\ {= {{\int_{0}^{T_{\alpha}\text{/}2}{{\frac{2}{\sqrt{3}} \cdot \frac{1}{k}}\left( {I_{L\; 0} + i_{L\; 6} + {\frac{2u_{L\; 1}}{L_{o}}t}} \right)e^{j{({{{\alpha\pi}\text{/}3} - {\pi \text{/}6}})}}{dt}}} + {\int_{0}^{T_{\beta}\text{/}2}{{\frac{2}{\sqrt{3}} \cdot \frac{1}{k}}\left( {I_{L\; 2} + i_{L\; 4} + {\frac{2u_{L\; 2}}{L_{o}}t}} \right)e^{j{({{{\beta\pi}\text{/}3} - {\pi \text{/}6}})}}{dt}}}}} \\ {= {{{\frac{2}{\sqrt{3}} \cdot \frac{1}{k}}\left( {{\left( {I_{L\; 0} + I_{L\; 6}} \right)\frac{T_{\alpha}}{2}} + {\frac{u_{L\; 1}}{L_{o}}\left( \frac{T_{\alpha}}{2} \right)^{2}}} \right)e^{j{({{{\alpha\pi}\text{/}3} - {\pi \text{/}6}})}}} + {{\frac{2}{\sqrt{3}} \cdot \frac{1}{k}}\left( {{\left( {I_{L\; 2} + I_{L\; 4}} \right)\frac{T_{\beta}}{2}} + {\frac{u_{L\; 2}}{L_{o}}\left( \frac{T_{\beta}}{2} \right)^{2}}} \right)e^{j{({{{\beta\pi}\text{/}3} - {\pi \text{/}6}})}}}}} \end{matrix} & (21) \end{matrix}$

where (α, β) can be (1,2) or (2,3) or (3,4) or (5,6) or (6,1), depending on which sector {right arrow over (I)}_(ref) is located in. For example, if {right arrow over (I)}_(ref) is located in sector I, (α, β) will be (1,2).

Substituting {right arrow over (I)}_(ref)=I_(ref)e^(jθ) into equation (21), the dwell times can be calculated under the following three different cases.

Case 1: when the inductance L_(o)>∞ or the inductance L_(o) is so large that the current ripple can be ignored so that i_(L0)=i_(L2)=i_(L4)=i_(L6)−I_(L), then the dwell times are the same as the known SVM.

T _(α)=mT_(s) sin(π/6−θ)   (22)

T _(β) =mT _(s) sin(π/6+θ)   (23)

T ₀ =T _(s) −T _(α) −T _(β)  (24)

where the modulation index m is given by:

$\begin{matrix} {{m = {k\frac{I_{ref}}{I_{L}}}},} & (25) \end{matrix}$

and θ is the angle between the reference current {right arrow over (I)}_(ref) and the α-axis as shown in FIG. 7.

In this case, the improved SVM according to various preferred embodiments of the present invention is consistent with the known SVM.

Case 2: When the inductance L_(o) is very small or the load is very light, then the load-side can be in DCM mode. The dwell times are calculated as:

$\begin{matrix} {T_{\alpha} = {2\sqrt{\frac{{kI}_{ref}L_{o}T_{s}{\sin \left( {{\pi \text{/}6} - \theta} \right)}}{{u_{1\; \alpha}\text{/}k} - u_{o}}}}} & (26) \\ {T_{\beta} = {2\sqrt{\frac{{kL}_{o}I_{ref}T_{s}{\sin \left( {{\pi \text{/}6} + \theta} \right)}}{{\beta_{1\; \beta}\text{/}k} - u_{o}}}}} & (27) \\ {T_{0} = {T_{s} - T_{\alpha} - T_{\beta}}} & (28) \end{matrix}$

where k is the transformer turns ratio, L_(o) is the inductance of the load-side inductor L₀, {right arrow over (I)}_(ref) is the magnitude of the vector {right arrow over (I)}_(ref) and is determined by the controller, T_(s) is the sampling period, θ is the angle between the reference current {right arrow over (I)}_(ref) and the a-axis as shown in FIG. 7, u_(1α) is measured by the controller and corresponds to a line-to-line voltage depending on the switching state, u_(1β) is measured by the controller and corresponds to a line-to-line voltage depending on the switching state, and u_(o) is the output voltage as measured by the controller. The line-to-line voltages u_(1α) and u_(1β) depend on the switching state. For example, in Sector I with active vectors {right arrow over (I)}₁ and {right arrow over (I)}₂, line-to-line voltages u_(1α) and u_(1β) are u_(ab) and u_(ac), respectively.

Case 3: when in CCM operation and the current ripple cannot be ignored, then the dwell times are calculated as:

$\begin{matrix} {T_{\alpha} = \frac{{- B} + \sqrt{B^{2} + {{AC}\mspace{14mu} {\sin \left( {{\pi \text{/}6} - \theta} \right)}}}}{A}} & (29) \\ {T_{\beta} = {\frac{{- B} + \sqrt{B^{2} + {{AC}\mspace{14mu} {\sin \left( {{\pi \text{/}6} - \theta} \right)}}}}{A} \cdot \frac{\sin \left( {{\pi \text{/}6} + \theta} \right)}{\sin \left( {{\pi \text{/}6} - \theta} \right)}}} & (30) \\ {{T_{0} = {T_{s} - T_{\alpha} - T_{\beta}}}{where}} & (31) \\ {A = {\left( {{4u_{1\alpha}\text{/}k} - u_{o}} \right) + {\left( {{4u_{1\beta}\text{/}k} - u_{o}} \right)\frac{\sin \left( {{\pi \text{/}6} + \theta} \right)}{\sin \left( {{\pi \text{/}6} - \theta} \right)}}}} & (32) \\ {B = {{2{nL}_{o}I_{L\; 0}} - {3u_{o}T_{s}\text{/}2}}} & (33) \\ {C = {2n^{2}{kLI}_{ref}T_{s}}} & (34) \end{matrix}$

where um is measured by the controller and corresponds to a line-to-line voltage depending on the switching state, u_(1β) is measured by the controller and corresponds to a line-to-line voltage depending on the switching state, k is the transformer turns ratio, u_(o) is the output voltage as measured by the controller, θ is the angle between the reference current {right arrow over (I)}_(ref) and the α-axis as shown in FIG. 7, L_(o) is the inductance of the load-side inductor L_(o), I_(L0) is the current through inductor L_(o) as measured by the controller at the beginning of the sampling period T_(s), T_(s) is the sampling period, and I_(ref) is the magnitude of the vector {right arrow over (I)}_(ref) and is determined by the controller. In one sampling period T_(s), the vector I_(α) is divided to n equal parts. In this example, n is 2 because one sampling period includes I_(α) and −I_(α). If n=2, then B and C are provided by:

B=4L _(o) I _(L0)3u _(o) T _(s)/2   (35)

C=8kL_(o)I_(ref)T_(s)   (36)

FIGS. 9A, 9C, and 9E show waveforms of the isolated matrix rectifier shown in FIG. 1 in CCM using known SVM, and FIGS. 9B, 9D, and 9F show corresponding waveforms of the isolated matrix rectifier shown in FIG. 1 in CCM using SVM according to various preferred embodiments of the present invention. In FIGS. 9A and 9B, the load-side inductor current is continuous, so the isolated matrix rectifier is operating in CCM. FIGS. 9C and 9D show the waveforms in the time domain, and FIGS. 9E and 9F show the waveforms in the frequency domain. Comparing these figures demonstrates that the improved SVM according to various preferred embodiments of the present invention provide a line-side current with a better shaped waveform and with a smaller THD. The THD using the improved SVM was measured as 4.71% while the THD using the known SVM was measured as 7.59%, for example.

FIGS. 10A, 10C, and 10E show waveforms of the isolated matrix rectifier shown in FIG. 1 in DCM using known SVM, and FIGS. 1013, 10D, and 10F show corresponding waveforms of the isolated matrix rectifier shown in FIG. 1 in DCM using SVM according to various preferred embodiments of the present invention. In FIGS. 10A and 10, the load-side inductor current is discontinuous (i.e., the current is equal to zero), so the isolated matrix rectifier is operating in DCM. FIGS. 10C and 10D show the waveforms in time domain, and FIGS. 10E and 10F show the waveforms in the frequency domain. Comparing these figures demonstrates that the improved SVM according to various preferred embodiments of the present invention provide a line-side current with a better shaped waveform and with a smaller THD. The THD using the improved SVM was measured as 6.81% while the THD using known SVM was measured as 17.4%, for example.

Thus, the improved SVM according to various preferred embodiments of the present invention is capable of being used with the isolated matrix rectifier in FIG. 1 in both CCM and DCM operation. The line-side current THD is significantly reduced with the improved SVM compared to known SVM. The improved SVM is suitable for the compact and high-efficiency design with a wide-load range. The improved SVM can also be applied to current-source converter to improve the AC side current THD.

In the preferred embodiments of the present, to calculate the dwell times, the controller measures transformer primary current i_(p) (or inductor current I_(L)), line voltages u_(a), u_(b), u_(c), and output voltage u_(o). The controller can be any suitable controller, including, for example, a PI controller, a PID controller, etc. The controller can be implemented in an IC device or a microprocessor that is programmed to provide the functions discussed above.

The same techniques and principles applied to the isolated matric rectifier in FIG. 1 can also be applied to the current-source rectifier in FIG. 2 and to the current-source inverter in FIG. 3. These techniques and principles are not limited to the devices shown in FIGS. 1-3 and can be applied to other suitable devices, including, for example, non-isolated devices.

It should be understood that the foregoing description is only illustrative of the present invention. Various alternatives and modifications can be devised by those skilled in the art without departing from the present invention. Accordingly, the present invention is intended to embrace all such alternatives, modifications, and variances that fall within the scope of the appended claims. 

What is claimed is:
 1. A converter comprising: a transformer including primary windings and secondary windings; switches connected to the primary windings; an output inductor connected to the secondary windings; and a controller connected to the switches; wherein the controller turns the switches on and off based on dwell times calculated using space vector modulation with a reference current {right arrow over (I)}_(ref) whose magnitude changes with time.
 2. A converter of claim 1, wherein: the switches include six switches; the space vector modulation includes using six active switching states and three zero switching states; a current space is divided into six sectors by the six active switching states such that a vector with θ=0 is located halfway between two of the active switching states; and magnitudes of the six active switching states change with time.
 3. A converter of claim 2, wherein: the controller turns the six switches on and off based on dwell times that are calculated based on an ampere-second balance equation: {right arrow over (I)}_(ref) T _(s)=∫₀ ^(T) ^(α) {right arrow over (I)} _(α) dt+∫ ₀ ^(T) ^(β) {right arrow over (I)} _(β) dt+∫ ₀ ^(T) ⁰ {right arrow over (I)} ₀ dt where {right arrow over (I)}_(ref)=I_(ref)e^(jθ), θ is an angle between the reference current {right arrow over (I)}_(ref) and the vector with θ=0, T_(s) is a sampling period, {right arrow over (I)}_(α), {right arrow over (I)}_(β), {right arrow over (I)}₀ are three nearest adjacent active vectors to {right arrow over (I)}_(ref), and T_(α),T_(β),T₀ are dwell times of {right arrow over (I)}_(α), {right arrow over (I)}_(β), {right arrow over (I)}₀.
 4. A converter of claim 3, wherein the controller turns the six switches on and off based on a vector sequence {right arrow over (I)}_(α), {right arrow over (I)}₀, −{right arrow over (I)}_(β), {right arrow over (I)}₀, {right arrow over (I)}_(β), {right arrow over (I)}_(β), {right arrow over (I)}₀, −_(α),{right arrow over (I)}₀, during the sampling period T_(s).
 5. A converter of claim 4, wherein the controller turns the six switches on and off based on a timing sequence T_(α)/2, T₀/4,T_(β)/2,T₀/4, T_(β)/2, T₀/4,T_(α)/2,T₀/4, during the sampling period T_(s).
 6. A converter of claim 4, wherein: the controller calculates the dwell times using: $T_{\alpha} = \frac{{- B} + \sqrt{B^{2} + {{AC}\mspace{14mu} {\sin \left( {{\pi \text{/}6} - \theta} \right)}}}}{A}$ $T_{\beta} = {\frac{{- B} + \sqrt{B^{2} + {{AC}\mspace{14mu} {\sin \left( {{\pi \text{/}6} - \theta} \right)}}}}{A} \cdot \frac{\sin \left( {{\pi \text{/}6} + \theta} \right)}{\sin \left( {{\pi \text{/}6} - \theta} \right)}}$ T₀ = T_(s) − T_(α) − T_(β) where $A = {\left( {{4u_{1\alpha}\text{/}k} - u_{o}} \right) + {\left( {{4u_{1\beta}\text{/}k} - u_{o}} \right)\frac{\sin \left( {{\pi \text{/}6} + \theta} \right)}{\sin \left( {{\pi \text{/}6} - \theta} \right)}}}$ B = 4L_(o)I_(L 0) − 3u_(o)T_(s)/2 C = 8kL_(o)I_(ref)T_(s) u_(1α) is a line-to-line voltage depending on the active switching state {right arrow over (I)}_(α), u_(1β) is a line-to-line voltage depending on the active switching state {right arrow over (I)}_(β), k is a transformer turns ratio, u₀ is an output voltage of the converter, θ is the angle between the reference current {right arrow over (I)}_(ref) and the vector with θ=0, L_(o) is an inductance of the output inductor, I_(L0) is the current through inductor L_(o) at a beginning of the sampling period T_(s), T_(s). is the sampling period, and I_(ref) is a magnitude of the vector {right arrow over (I)}_(ref).
 7. A converter of claim 4, wherein: the controller calculates the dwell times using: $T_{\alpha} = {2\sqrt{\frac{{kI}_{ref}L_{o}T_{s}{\sin \left( {{\pi \text{/}6} - \theta} \right)}}{{u_{1\; \alpha}\text{/}k} - u_{o}}}}$ $T_{\beta} = {2\sqrt{\frac{{kL}_{o}I_{ref}T_{s}{\sin \left( {{\pi \text{/}6} - \theta} \right)}}{{u_{1\; \beta}\text{/}k} - u_{o}}}}$ where k is a transformer turns ratio, L_(o) is an inductance of the output inductor, I_(ref) is the magnitude of the vector {right arrow over (I)}_(ref), T_(s) is the sampling period, θ is an angle between the reference current {right arrow over (I)}_(ref) and the vector with θ=0, u_(1α) is a line-to-line voltage depending on the active switching state {right arrow over (I)}_(α), u_(1β) is a line-to-line voltage depending on the active switching state {right arrow over (I)}_(β), and u_(o) is an output voltage of the converter.
 8. A converter of claim 1, wherein the converter is one of a matrix rectifier, a current-source rectifier, and a current-source inverter.
 9. A converter of claim 1, wherein the converter is operated in a continuous-conduction mode.
 10. A converter of claim 1, wherein the converter is operated in a discontinuous-conduction mode.
 11. A space-vector-modulation method for a converter including a transformer with primary windings and secondary windings, switches connected to the primary windings, and an output inductor connected to the secondary windings, the space-vector-modulation method comprising: turning the switches on and off based on dwell times calculated using space vector modulation with a reference current {right arrow over (I)}_(ref) whose magnitude changes with time.
 12. A method of claim 11, wherein: the switches include six switches; calculating the dwell times uses: six active switching states and three zero switching states; and a current space that is divided into six sectors by the six active switching states such that a vector with θ=0 is located halfway between two of the active switching states; and magnitudes of the six active switching states change with time.
 13. A method of claim 12, wherein: turning the six switches on and off is based on dwell times that are calculated based on an ampere-second balance equation: {right arrow over (I)}_(ref) T _(s)=∫₀ ^(T) ^(α) {right arrow over (I)} _(α) dt+∫ ₀ ^(T) ^(β) {right arrow over (I)} _(β) dt+∫ ₀ ^(T) ⁰ {right arrow over (I)} ₀ dt where {right arrow over (I)}_(ref)=I_(ref)e^(jθ), θ is an angle between the reference current {right arrow over (I)}_(ref) and the vector with θ=0, T_(s) is a sampling period, {right arrow over (I)}_(α), {right arrow over (I)}_(β), {right arrow over (I)}₀ are three nearest adjacent active vectors to {right arrow over (I)}_(ref) , and T_(α),T_(β),T₀ are dwell times of {right arrow over (I)}_(α), {right arrow over (I)}_(β), {right arrow over (I)}₀.
 14. A method of claim 13, wherein turning the six switches on and off is based on a vector sequence {right arrow over (I)}_(α), {right arrow over (I)}₀, −{right arrow over (I)}_(β), {right arrow over (I)}₀, {right arrow over (I)}_(β), {right arrow over (I)}₀, −{right arrow over (I)}_(α), {right arrow over (I)}₀, during the sampling period T_(s).
 15. A method of claim 14, wherein turning the six switches on and off is based on a timing sequence T_(α)/2, T₀/4,T_(β)/2,T₀/4, T_(β)/2, T₀/4,T/2,T₀/4, during the sampling period T_(s).
 16. A method of claim 14, wherein: the dwell times are calculated using: $T_{\alpha} = \frac{{- B} + \sqrt{B^{2} + {{AC}\mspace{14mu} {\sin \left( {{\pi \text{/}6} - \theta} \right)}}}}{A}$ $T_{\beta} = {\frac{{- B} + \sqrt{B^{2} + {{AC}\mspace{14mu} {\sin \left( {{\pi \text{/}6} - \theta} \right)}}}}{A} \cdot \frac{\sin \left( {{\pi \text{/}6} + \theta} \right)}{\sin \left( {{\pi \text{/}6} - \theta} \right)}}$ T₀ = T_(s) − T_(α) − T_(β) where $A = {\left( {{4u_{1\alpha}\text{/}k} - u_{o}} \right) + {\left( {{4u_{1\beta}\text{/}k} - u_{o}} \right)\frac{\sin \left( {{\pi \text{/}6} + \theta} \right)}{\sin \left( {{\pi \text{/}6} - \theta} \right)}}}$ B = 4L_(o)I_(L 0) − 3u_(o)T_(s)/2 C = 8kL_(o)I_(ref)T_(s) u_(1α) is a line-to-line voltage depending on the active switching state {right arrow over (I)}_(α), u_(1β) is a line-to-line voltage depending on the active switching state {right arrow over (I)}_(β), k is a transformer turns ratio, u_(o) is an output voltage of the converter, θ is the angle between the reference current {right arrow over (I)}_(ref) and the vector with θ=0, L_(o) is an inductance of the output inductor, I_(L0) is the current through inductor L_(o) at a beginning of the sampling period T_(s), T_(s) is the sampling period, and I_(ref) is a magnitude of the vector {right arrow over (I)}_(ref).
 17. A method of claim 14, wherein: the controller calculates the dwell times using: $T_{\alpha} = {2\sqrt{\frac{{kI}_{ref}L_{o}T_{s}{\sin \left( {{\pi \text{/}6} - \theta} \right)}}{{u_{1\; \alpha}\text{/}k} - u_{o}}}}$ $T_{\beta} = {2\sqrt{\frac{{kL}_{o}I_{ref}T_{s}{\sin \left( {{\pi \text{/}6} - \theta} \right)}}{{u_{1\; \beta}\text{/}k} - u_{o}}}}$ where k is a transformer turns ratio, L_(o) is an inductance of the output inductor, I_(ref) is the magnitude of the vector {right arrow over (I)}_(ref), T_(s). is the sampling period, θ is an angle between the reference current {right arrow over (I)}_(ref) and the vector with θ=0, u_(1α) is a line-to-line voltage depending on the active switching state {right arrow over (I)}_(α), u_(1β) is a line-to-line voltage depending on the active switching state {right arrow over (I)}_(β), and u_(o) is an output voltage of the converter.
 18. A method of claim 11, wherein the converter is one of a matrix rectifier, a current-source rectifier, and a current-source inverter.
 19. A method of claim 11, further comprising operating the converter in a continuous-conduction mode.
 20. A method of claim 11, further comprising operating the converter in a discontinuous-conduction mode. 